General three and four person two color Hat Game

distinguishable players are randomly fitted with a white or black hat, where the probabilities of getting a white or black hat may be different for each player, but known to all the players. All players guess simultaneously the color of their own hat observing only the hat colors of the other − 1 players. It is also allowed for each player to pass: no color is guessed. The team wins if at least one player guesses his hat color correctly and none of the players has an incorrect guess. No communication of any sort is allowed, except for an initial strategy session before the game begins. Our goal is to maximize the probability of winning the game and to describe winning strategies, using the concept of an adequate set. We find explicit solutions in case of =3 and =4.